Question

I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection $p_i:X\times X\rightarrow X$ for $i=1,2$. I would like to understand the following statement.

The scheme of isomorphisms $Isom(p_1^*E,p_2^*E)$ is a principal $GL(n)$-bundle over $X\times X$. We define the symmetry groupoid of $E\rightarrow X$ to be $Isom(p_1^*E,p_2^*E) \rightrightarrows X$.

As a trivial example, take $X=Spec(k)$ for a field $k$. Then the vector bundle $E$ is a simply $k$-vector space and $Isom(p_1^*E,p_2^*E) \cong GL(n,k)$. So it makes sense to make a such definition. For a general $X$, the definition above is hard for me to digest. My questions are:

1. Why do we need $Isom(p_1^*E,p_2^*E)$, instead of $Isom(E,E)$?
2. What is the two maps $Isom(p_1^*E,p_2^*E) \rightrightarrows X$?
3. Why is it reasonable to call $Isom(p_1^*E,p_2^*E) \rightrightarrows X$ the symmetry groupoid of $E\rightarrow X$?

I would appreciate it if some experts could help me understanding these things.

Answer 1

I'm not an algebraic geometer so let's call $X$ a real manifold. I don't think that really matters it could be a topological space or even a set for what I am about to say. Assume that $E \to X$ is a rank $n$ real vector bundle. For any $x \in X$ let $E_x$ be the fibre of $E$ over $x$. The natural structure you have in this situation is that if $f \colon E_{x} \to E_{y} $ and $g \colon E_{y} \to E_{z}$ are isomorphisms then you can compose to get $g \circ f \colon E_{x} \to E_{z}$ also an isomorphism. From this follows the fact that you have a groupoid whose objects are all $x \in X$ and whose morphisms from $x$ to $y$ are $Isom(E_x, E_y)$ (or $Isom(E_y, E_x)$ depending on how you like to compose morphisms.)

(1) $Isom(E, E)$ is a perfectly reasonable object, it's a bundle of groups over $X$. But it doesn't capture all the information such as isomorphisms from $E_x$ to $E_y$ where $x \neq y$.

(2) $Isom(p_1^*E, p_2^*E) $ is the union over all $x, y \in X$ of $Isom(E_x, E_y)$ and if $f \in Isom(E_x, E_y)$ then the two maps are $f \mapsto x$ and $f \mapsto y)$.

(3) I am not sure of the answer to this but it seems reasonable to me that this groupoid captures information about the symmetries of $E \to X$.

I don't think that $Isom(p_1^*E, p_2^*E) $ being a principal $GL(n, \mathbb{R})$ bundle is correct. I don't see any reason why $Isom(E_x, E_y)$, for example, is acted on by $GL(n, \mathbb{R})$.